Probability Review Thinh Nguyen Probability Theory Review n
Probability Review Thinh Nguyen
Probability Theory Review n n n Sample space Bayes’ Rule Independence Expectation Distributions
Sample Space - Events n Sample Point q n Sample Space S q q n The outcome of a random experiment The set of all possible outcomes Discrete and Continuous Events q q A set of outcomes, thus a subset of S Certain, Impossible and Elementary
Set Operations n Union Intersection Complement n Properties n n q Commutation q Associativity q Distribution q De Morgan’s Rule S
Axioms and Corollaries n Axioms n n n If A 1, A 2, … are pairwise exclusive Corollaries
Conditional Probability n Conditional Probability of event A given that event B has occurred n If B 1, B 2, …, Bn a partition of S, then S B 1 B 2 A (Law of Total Probability) B 3
Bayes’ Rule n If B 1, …, Bn a partition of S then
Event Independence n Events A and B are independent if n If two events have non-zero probability and are mutually exclusive, then they cannot be independent
Random Variables
Random Variables n The Notion of a Random Variable q q n The outcome is not always a number Assign a numerical value to the outcome of the experiment Definition q A function X which assigns a real number X(ζ) to each outcome ζ in the sample space of a random experiment S ζ X(ζ) = x x Sx
Cumulative Distribution Function n Defined as the probability of the event {X≤x} Fx(x) 1 n Properties x Fx(x) 1 ¾ ½ ¼ 0 1 2 3 x
Types of Random Variables n Continuous q Probability Density Function n Discrete q Probability Mass Function
Probability Density Function n The pdf is computed from n Properties f. X(x) dx n For discrete r. v. x
Expected Value and Variance n n The expected value or mean of X is n The variance of X is n The standard deviation of X is n Properties
Queuing Theory
Example n. Send a file over the internet packet Modem card buffer link (fixed rate)
place C Computation (Queuing) transmission propagation Delay Models B A time
Queue Model
Practical Example
Multiserver queue
Multiple Single-server queues
Standard Deviation impact
Queueing Time
Queuing Theory The theoretical study of waiting lines, expressed in mathematical terms input server queue Delay= queue time +service time output
The Problem Given n One or more servers that render the service n A (possibly infinite) pool of customers n Some description of the arrival and service processes. Describe the dynamics of the system Evaluate its Performance n If there is more than one queue for the server(s), there may also be some policy regarding queue changes for the customers.
Common Assumptions n n The queue is FCFS (FIFO). We look at steady state : after the system has started up and things have settled down. State=a vector indicating the total # of customers in each queue at a particular time instant (all the information necessary to completely describe the system)
Notation for queuing systems nomitted : Where A and B can be D for Deterministic distribution M for Markovian (exponential) distribution G for General (arbitrary) distribution if infinite
The M/M/1 System Poisson Process Exponential server queue output
Arrivals follow a Poisson process n. Readily amenable for analysis n. Reasonable n n n for a wide variety of situations a(t) = # of arrivals in time interval [0, t] = mean arrival rate t = k ; k = 0, 1, …. ; 0 Pr(exactly 1 arrival in [t, t+ ]) = Pr(no arrivals in [t, t+ ]) = 1 - Pr(more than 1 arrival in [t, t+ ]) = 0 Pr(a(t) = n) = e- t ( t)n/n!
Model for Interarrivals and Service times · · · Customers arrive at times t 0 < t 1 <. . - Poisson distributed The differences between consecutive arrivals are the interarrival times : n = tn - t n-1 n in Poisson process with mean arrival rate exponentially distributed, , are Pr( n t) = 1 - e- t n Service times are exponentially distributed, with mean service rate : Pr(Sn s) = 1 - e- s
System Features n Service times are independent service times are independent of the arrivals n Both inter-arrival and service times are memoryless n Pr(Tn > t 0+t | Tn> t 0) = Pr(Tn t) future events depend only on the present state This is a Markovian System
Exponential Distribution
Markov Models Buffer Occupancy • n+1 departure • n-1 • n arrival
Probability of being in state n
Steady State Analysis
Markov Chains 0 1 . . . n-1 n n+1
Substituting Utilization
Substituting P 1 • Higher states have decreasing probability • Higher utilization causes higher probability of higher states
What about P 0 Queue determined by
E(n), Average Queue Size
Selecting Buffers For large utilization, buffers grow exponentially
Throughput n Throughput=utilization/service time = /Ts n For =. 5 and Ts=1 ms n Throughput is 500 packets/sec
Intuition on Little’s Law n If a typical customer spends T time units, on the overage, in the system, then the number of customers left behind by that typical customer is equal to
Applying Little’s Law
Probability of Overflow
Buffer with N Packets
Example n Given q q n Arrival rate of 1000 packets/sec Service rate of 1100 packets/sec Find q Utilization q Probability of having 4 packets in the queue
Example
Application to Statistcal Multiplexing n n Consider one transmission line with rate R. Time-division Multiplexing q n Divide the capacity of the transmitter into N channels, each with rate R/N R/N Statistical Multiplexing q Buffering the packets coming from N streams into a single buffer and transmitting them one at a time. R
Network of M/M/1 Queues
M/G/1 Queue The customers pat at rate since each customer pays on the average and customers go through the queue per unit time. Assume that every customer in the queue pays at rate R when his or her remaining service time is equal R. time t, the customers pay at a rate At atogiven equal to the sum of the remaining service times of all the customer in the queue. The queue begin first come-first served, this sum is equal to Total cost paid by a customer: the queueing time of a customer who would enter the queue at time t. Expected cost paid by each customer: S 0 Q S
- Slides: 51